The Annals of Applied Probability

Weighted multilevel Langevin simulation of invariant measures

Gilles Pagès and Fabien Panloup

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We investigate a weighted multilevel Richardson–Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions adapted from the one introduced in [Bernoulli 23 (2017) 2643–2692] for regular Monte Carlo simulation. In a first result, we prove under weak confluence assumptions on the diffusion, that for any integer $R\ge2$, the procedure allows us to attain a rate $n^{\frac{R}{2R+1}}$ whereas the original algorithm convergence is at a weak rate $n^{1/3}$. Furthermore, this is achieved without any explosion of the asymptotic variance. In a second part, under stronger confluence assumptions and with the help of some second-order expansions of the asymptotic error, we go deeper in the study by optimizing the choice of the parameters involved by the method. In particular, for a given $\varepsilon>0$, we exhibit some semi-explicit parameters for which the number of iterations of the Euler scheme required to attain a mean-squared error lower than $\varepsilon^{2}$ is about $\varepsilon^{-2}\log(\varepsilon^{-1})$.

Finally, we numerically test this multilevel Langevin estimator on several examples including the simple one-dimensional Ornstein–Uhlenbeck process but also a high dimensional diffusion motivated by a statistical problem. These examples confirm the theoretical efficiency of the method.

Article information

Ann. Appl. Probab., Volume 28, Number 6 (2018), 3358-3417.

Received: July 2016
Revised: June 2017
First available in Project Euclid: 8 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy) 65C05: Monte Carlo methods

Ergodic diffusion invariant measure multilevel ergodicity Richardson–Romberg Monte Carlo PAC-Bayesian


Pagès, Gilles; Panloup, Fabien. Weighted multilevel Langevin simulation of invariant measures. Ann. Appl. Probab. 28 (2018), no. 6, 3358--3417. doi:10.1214/17-AAP1364.

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Supplemental materials

  • Supplement to “Weighted multilevel Langevin simulation of invariant measures”. In order to improve the readability of the current article, several technical proofs have been postponed in a supplementary document. In the case in point, the precise reference is given at the end of the proposition.